Word length asymptotics for actions of hyperbolic groups via stationary measures for Markov processe

Group Actions and Dynamics
Event time: 
Monday, January 23, 2017 - 11:30am to 12:30pm
205 LOM
Ilya Gekhtman
Speaker affiliation: 
Yale University
Event description: 

Abstract: Consider any nonelementary action of a hyperbolic group G on a not necessarily proper Gromov hyperbolic space X. The action is not assumed to be discrete (for example, it could be a dense subgroup of SL_{2}\R) and X is not assumed to be proper (for example it could be the curve complex, on which the mapping class group acts with pseudo-Anosov elements acting as loxodromics).

We prove certain asymptotic properties for the action, including the following.
1)With respect to the Patterson-Sullivan measure on the boundary of G, the image in X of almost every word-geodesic in G sublinearly tracks a geodesic in X.
2)The proportion of elements in a Cayley-ball of radius R in G which act loxodromically on X converges to 1 with R.

A major tool is Cannon’s theorem that hyperbolic groups admit geodesic automation.

I will also discuss applications to other Markov processes on groups such as non-backtracking random walks and ideas for extending these methods beyond hyperbolic groups. This is based on completed and ongoing work with Sam Taylor and Giulio Tiozzo.